Both eyes open according to Nick Penn of pennsport.
I've just listened to part of it. Quickly, I got up to coming from below or behind a rising or driven target, with both eyes open "you'll never loose sight of it". Rubbish.....That depends on the amount of lead it requires as at some point, the gun and your hand placed on the forend will block view from both eyes. I've just generated images showing at what stage a trap shooter will loose view of the apron and target release point when holding a distance above the trench. This is effected by the way in which your hand is placed on the forend, the size of the forend, the distance from your eyes to the same component, the distance between an individuals eyes. This can all be easily demonstrated and proven. It is the same effect as seen on springing teal or straight overhead driven targets that require more lead that vision from the non-aligned eye will allow.
"If you pull the trigger when it feels righ, you'll never miss another clay".....

....... don't believe that, there are many other contributing factiors to success, but I like the positiveness
You say that "you'd not like to shoot off your weak side"? It's only your weak side because you've not trained it. Someone with a dominant left eye should shoot from the same shoulder, but if not pracical, too late in life, have a RH gun with a palm swell you like etc etc. continue to shoot with both eyes open, but train to dim the cross dominant eye just before the point where you contact the target. It's not perfect, but it will improve things. Ensure there's no head tilting, allowing full vision of the target from your "shooting eye", and learn how to rely on the best part of your vision, and trasition from peripheral to central vision at the appropriate time, hence making best use of the part of vision that is best at tracking movement, speed and direction. Once you fixate with central vision on a target, you will loose these attributes. Your central vision is designed for detail, colour and distance judgement.
Yes get a coach or even an instructor, but ensure its a good one that is fully up to speed on the vision side of "eye"- hand coordination sports

A good one will be able to teach and bring to your attention all the important aspects of vision in moving target sports that involve pointing rather than aiming. Had someone say today that he does not believe anyone that "never" or "do not see their barrels". Of course he is right, it's just the degree of consciousness that goes into seeing or not seeing what is directly infront of your eyes (both if it's both eyes open). Like most sports, it very much comes down to common sense and not overcomplicating what in essence, is a relatively simple sporting action
I may listen to the rest tomorrow
